From (REF ), \(U_{\varepsilon }=\frac{2\sqrt{2}\varepsilon }{\varepsilon ^2+|x|^2}\) , taking a radially decreasing cut-off function \(\xi \in C_0^{\infty }(\mathbb {R}^4)\)  such that \(\xi \equiv 1\)  in \(B_1\) , \(\xi \equiv 0\)  in \(\mathbb {R}^4 \backslash B_2\) , and let \(W_{\varepsilon }(x) = \xi (x) U_{\varepsilon }(x)\) . We have (see [1]}),
\(\Vert \nabla W_\varepsilon \Vert ^2_2=\mathcal {S}^2+O(\varepsilon ^2), \quad \quad \Vert W_\varepsilon \Vert ^4_4=\mathcal {S}^2+O(\varepsilon ^4),\) 
